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Math Help - complex bilinear extension

  1. #1
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    complex bilinear extension

    Hello,

    I came across the following statement:

    "..., where g is the complex bilinear extension of the standard scalar product on R^n to C^n (note that this a symmetric form, not a Hermitian one)."

    Can someone tell me how the "complex bilinear extension" is defined, i.e. if x,y in C^n, then g(x,y)=... .

    Thanks in advance!
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  2. #2
    MHF Contributor

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    Re: complex bilinear extension

    The "standard" scalar product on R^n is \sqrt{xy} where x and y are vectors in R^n. If x and y are vectors in C^n, the scalar product is \sqrt{xy^*} where "y*" indicates the complex conjugate of y: if y= (a_1+ ib_1, a_2+ ib_2, ..., a_n+ ib_n) then y^*= (a_1- ib_1, a_2- ib_2, ..., a_n- ib_n). That is, if x= (c_1+ id_1, c_2+ id_2, ..., c_n+ id_n) and y= (a_1+ ib_1, a_2+ ib_2, ..., a_n+ ib_n) then the inner product of x and y is (c_1+ id_1)(a_1- ib_1)+ (c_2+ id_2)(a_2- ib_2)+ ...+ (c_n+ id_n)(a_1- ib_n).
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  3. #3
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    Re: complex bilinear extension

    I know that definition, but that's not symmetric or bilinear, so I don't think thats what's meant by "complex bilinear extension", which also should be symmetric.

    Maybe it's what you told me, just without the conjugate.
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